Tetrachord: Difference between revisions
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John Chalmers, in [http://eamusic.dartmouth.edu/%7Elarry/published_articles/divisions_of_the_tetrachord/index.html Divisions of the Tetrachord], tells us: | John Chalmers, in [http://eamusic.dartmouth.edu/%7Elarry/published_articles/divisions_of_the_tetrachord/index.html Divisions of the Tetrachord], tells us: | ||
''Tetrachords are modules from which more complex scalar and harmonic structures may be built. These structures range from the simple heptatonic scales known to the classical civilizations of the eastern Mediterranean to experimental gamuts with many tones. Furthermore, the traditional scales of much of the world's music, including that of Europe, the [[Arabic, | ''Tetrachords are modules from which more complex scalar and harmonic structures may be built. These structures range from the simple heptatonic scales known to the classical civilizations of the eastern Mediterranean to experimental gamuts with many tones. Furthermore, the traditional scales of much of the world's music, including that of Europe, the [[Arabic, Turkish, Persian|Near East]], the Catholic and Orthodox churches, Iran and India, are still based on tetrachords. Tetrachords are thus basic to an understanding of much of the world's music.'' | ||
Related pages: [[ | Related pages: [[22edo tetrachords]], [[17edo tetrachords]], [[Tricesimoprimal Tetrachordal Tesseract]], [[Armodue armonia#Creating scales with Armodue: modal systems-Modal systems based on tetrachords and pentachords|16edo tetrachords]], [[Gallery of Wakalixes#Divisions of the Tetrachord|Wakalix tetrachords]] | ||
__FORCETOC__ | __FORCETOC__ | ||
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=Ancient Greek Genera= | =Ancient Greek Genera= | ||
The ancient Greeks distinguished between three primary genera depending on the size of the largest interval, the characteristic interval, or | The ancient Greeks distinguished between three primary genera depending on the size of the largest interval, the characteristic interval, or CI—the enharmonic, chromatic, and diatonic. Modern theorists have added a fourth genera, called hyperenharmonic. | ||
===hyperenharmonic genus=== | ===hyperenharmonic genus=== | ||
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==Superparticular Intervals== | ==Superparticular Intervals== | ||
In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are [[ | In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are [[superparticular]]. | ||
=Ajnas (tetrachords in middle-eastern music)= | =Ajnas (tetrachords in middle-eastern music)= | ||
The concept of the tetrachord is extensively used in [[Arabic, | The concept of the tetrachord is extensively used in [[Arabic, Turkish, Persian|middle eastern]] music theory. The Arabic word for tetrachord is "jins" (singular form) or "ajnas" (plural form). | ||
See [http://www.maqamworld.com/ajnas.html maqamworld.com] for details. | See [http://www.maqamworld.com/ajnas.html maqamworld.com] for details. | ||
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[tetrachord], 9/8, [tetrachord] | [tetrachord], 9/8, [tetrachord] | ||
Of course, a tetrachord doesn't need to be paired with its copy. You might pair it with a dissimilar tetrachord ( | Of course, a tetrachord doesn't need to be paired with its copy. You might pair it with a dissimilar tetrachord (e.g. 1/1, c, d, 4/3): | ||
1/1, a, b, 4/3, 3/2, 3c/2, 3d/2, 2/1 | 1/1, a, b, 4/3, 3/2, 3c/2, 3d/2, 2/1 | ||
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ssL, sLs, Lss | ssL, sLs, Lss | ||
And, if you have only one step size (as is the case in [[ | And, if you have only one step size (as is the case in [[Porcupine]] temperament, for instance), you have an evenly-divided fourth, and only one possible rotation. (A tetrachord of this type can be found in [[22edo]] - see [[22edo tetrachords]].) | ||
=Tetrachords in equal temperaments= | =Tetrachords in equal temperaments= | ||
Naturally, any equally divided scale which contains an approximation of 4/3 will have its own family of tetrachords, starting with [[ | Naturally, any equally divided scale which contains an approximation of 4/3 will have its own family of tetrachords, starting with [[7edo]], which has one tetrachord: | ||
1 + 1 + 1 | 1 + 1 + 1 | ||
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|} | |} | ||
==Tetrachords of [[ | ==Tetrachords of [[10edo]]== | ||
Another example: 10edo has an interval that can function as a perfect fourth at 4 degrees, measuring 480 cents. It can thus be divided into any arrangement of two 1-degree steps and one 2-degree step: | Another example: 10edo has an interval that can function as a perfect fourth at 4 degrees, measuring 480 cents. It can thus be divided into any arrangement of two 1-degree steps and one 2-degree step: | ||
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Note that these tetrachords are all rotations of each other, and in terms of Greek genera, they are all "diatonic" (because the largest interval, or characteristic interval, at 240 cents, is less than 250 cents). | Note that these tetrachords are all rotations of each other, and in terms of Greek genera, they are all "diatonic" (because the largest interval, or characteristic interval, at 240 cents, is less than 250 cents). | ||
See also: [[ | See also: [[Armodue armonia#Creating scales with Armodue: modal systems-Modal systems based on tetrachords and pentachords|16edo tetrachords]], [[17edo tetrachords]], [[22edo tetrachords]], [[Tricesimoprimal Tetrachordal Tesseract]] (tetrachords of [[31edo]]). If you'd like to wrestle some tetrachords out of an equal temperament, please consider making a page for them and linking to that page from here! | ||
=Dividing Other-Than-Perfect Fourths= | =Dividing Other-Than-Perfect Fourths= | ||
A composer may choose to treat other-than-perfect fourths as material for constructing tetrachords. Some of the low-number equal temperaments contain diminished or augmented fourths, but nothing resembling a perfect fourth: [[ | A composer may choose to treat other-than-perfect fourths as material for constructing tetrachords. Some of the low-number equal temperaments contain diminished or augmented fourths, but nothing resembling a perfect fourth: [[6edo]], [[8edo]], [[9edo]], [[11edo]], [[13edo]], [[16edo]]. Also, one may divide a just other-than-perfect fourth, such as 21/16, 43/32, 26/19, 11/8. At what point does the concept of "tetrachord" stop being useful? | ||
=Tetrachords And Non-Octave Scales= | =Tetrachords And Non-Octave Scales= | ||
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Dividing a tenth into three equal parts generates a cycle of three fourths, they resembling perfect fourths when the division is done on a minor tenth. | Dividing a tenth into three equal parts generates a cycle of three fourths, they resembling perfect fourths when the division is done on a minor tenth. | ||
An example with [[ | An example with [[Carlos Gamma]]: | ||
[http://www.seraph.it/dep/det/GloriousGuitars.mp3 Glorious Guitars] by [[ | [http://www.seraph.it/dep/det/GloriousGuitars.mp3 Glorious Guitars] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/e8a36018d6b782c8ff7bc2416fa7ea5b-47.html blog entry]) | ||
[[Category: | [[Category:List]] | ||
[[Category: | [[Category:Scale]] | ||
[[Category: | [[Category:Tetrachord]] | ||
[[Category: | [[Category:Theory]] | ||